A graph isomorphism invariant based on neighborhood aggregation

TL;DR

This paper introduces a new graph isomorphism invariant called $rak{w}$-labeling, enabling polynomial-time algorithms to distinguish various graph classes, with practical applications in graph database searching.

Contribution

It proposes $rak{w}$-labeling and $rak{s}^k$-labeling as novel invariants, improving graph isomorphism testing and fingerprinting methods for complex graph classes.

Findings

Distinguishes all non-cospectral graph pairs.

Identifies all trees and 3-connected planar graphs.

Enables fast graph database searches.

Abstract

This paper presents a new graph isomorphism invariant, called $\mathfrak{w}$-labeling, that can be used to design a polynomial-time algorithm for solving the graph isomorphism problem for various graph classes. For example, all non-cospectral graph pairs are distinguished by the proposed combinatorial method, furthermore, even non-isomorphic cospectral graphs can be distinguished assuming certain properties of their eigenspaces. We also investigate a refinement of the aforementioned labeling, called $\mathfrak{s}^k$-labeling, which has both theoretical and practical applications. Among others, it can be used to generate graph fingerprints, which uniquely identify all graphs in the considered databases, including all strongly regular graphs on at most 64 nodes and all graphs on at most 12 nodes. It provably identifies all trees and 3-connected planar graphs up to isomorphism, which -- as a byproduct -- gives a new isomorphism algorithm for both graph classes. The practical importance of this fingerprint lies in significantly speeding up searching in graph databases, which is a commonly required task in biological and chemical applications.